5 COMPACT OPERATORS IN HUBERT SPACES 333

and by the uniqueness of that expression, we see that u(x) = 0 implies
x e F^; in other words, F^ = u~l(fyy which ends the proof of (11.5.7).

Remarks

(11.5.8) Let E0 be a prehilbert space which is a dense subspace of a Hilbert
space E (it can be proved that for any prehilbert space E0 there is a Hilbert
space E having that property; we have proved in (6.6.2) the special case of
that theorem in which E0 is separable). Let u be a compact self-adjoint
operator in E0; then the results (a), (b), and (c) of theorem (11.5.7) hold
without change for u. For it follows from the principle of extension of iden-
tities that the unique continuous extension u of u to E is self-adjoint, and it
is readily verified that \\u\\ = \\u\\; our assertion then follows from (11.4.2) and
from the following remark: if F0 is a finite dimensional subspace of E0, G0 its
orthogonal supplement in E0, G its orthogonal supplement in E, then G0 is
dense in G; this is a consequence of the fact that if v = 1E — PFo in ££(E)
(notations of Section 6.3), v is continuous and i?(E) = G, 0(E0) = G0 (see
(3.11.3)). With respect to the part (d) of (11.5.7), it is clear that the kernel of
u is the intersection of E0 with the kernel of u, hence is the subspace of vectors
of E0 orthogonal to all eigenspaces E(X) with X + 0. But if we consider the
canonical decomposition x = ]T x'k 4- £ x'i + *o °f an element x e E0, the

k k

sums on the right-hand side and the element x0 do not necessarily belong to E0.

(11.5.9) If x = £ x'k Hh £ xt + x0,y = ^y'k + 5>i' + y0 are the canonical

k k k k

decompositions 'of two vectors x, y of E, then

(«(*) I y) = E ttkte I yQ + Z vW I y®

k k

the series on the right-hand side being absolutely convergent (Section 6.4).
This formula at once shows that the self-adjoint operator u is positive if and
only if there are no negative eigenvalues vk, and that it is nondegenerate if
and only if u~l(Q) = {0}. If u is nondegenerate, and if in each eigenspace
E(A) (A 7^ 0), we take a Hilbert basis BA (consisting of a finite number of
vectors), then the union of the BA is a denumerable set which constitutes a
Hilbert basis in E (Section 6.5).

(11.5.10) Under the assumptions of (11.5.8), it should be observed that
it is quite possible that the self-adjoint compact operator u in E0 is non-
degenerate, whereas its continuous extension i7 to E is degenerate (in other
words, the kernel of u is not necessarily dense in the kernel of u); this may
happen even if u is a positive self-adjoint operator.) - w*(y/))| < 6, and either u*(y) = u*(yj) or we can take x = z/\\z\\,